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Math 137 Week 11 Notes – Dr. Paula Smith
Section 7.7 Approximate Integration
It is preferable to find the exact value of a definite integral, but there are many functions which
we will not be able to integrate, no matter what techniques we learn.
However, the value of any
definite integral can be approximated, not only by the Left and Righthand Riemann sums, but
also and more precisely by the Trapezoidal Rule, the Midpoint Rule, and Simpson’s Rule.
The Trapezoidal Rule for n, T
n
, approximates the area under the function on the interval by
means of a series of trapezoids, each with base of width
Δ
x = (b – a)/n and the height of both
sides, f(x
i1
) and f(x
i
), where x
i
= x
0
+ i
Δ
x.
We can either calculate both L
n
and R
n
, and then
average them for T
n
, or else we can use the formula
T
n
= (
Δ
x/2) [f(x
0
) + 2 f(x
1
) + 2 f(x
2
) + … + 2 f(x
n1
) + f(x
n
)]
The Midpoint Rule for n, M
n
, is a Riemann sum, where x
i
* is the midpoint of the ith segment.
So the formula is M
n
=
Δ
x [f(x
1
*) + f(x
2
*) + … + f(x
n1
*) + f(x
n
*)]
Experimentally, we see that T
n
and M
n
are each more precise than either L
n
or R
n
.
The question
is, how good are these rules? If the exact value of
∫
a
b
f(x) dx is I, then the error for the
Trapezoidal rule for n segments is E
T
= I – T
n
, and the error for the Midpoint rule for n segments
is E
M
= I – M
n
.
We will not know I, but by knowing bounds for the error, we can know how far
the true value I is from the error, or alternatively, we can construct an interval around the error
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This note was uploaded on 04/13/2010 for the course MATH 137 taught by Professor Speziale during the Spring '08 term at Waterloo.
 Spring '08
 SPEZIALE
 Math

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